The 54-year-old Australian, who was appointed as ATP executive chairman and president 12 months ago, will continue in his role until a successor is found.
"It has been a privilege to serve as executive chairman and president of the ATP, an organisation that I've been a part of for more than 35 years since I became a professional tennis player," Drewett said.
"I hold the ATP very close to my heart, and it's with sadness that I make the decision to enter this transition period due to my ill health."
In his brief time in charge, Drewett helped to convince the Grand Slam events to award the players more prize money and has been working actively to streamline the calendar.
Motor Neurone Disease, also down as ALS or Lou Gehrig's Disease, is a group of diseases which hit the nerve cells controlling the muscles.
World number two Roger Federer, the president of the ATP Player Council, said it was "sad news for all of us at the ATP and the entire tennis community".
"He is well liked and respected by everyone and has done a tremendous job in leading the ATP over the past 12 months, overseeing some major initiatives and a record-breaking year in 2012," the Swiss said.
"His dedication and service to the sport over the years has been truly admirable and he has been a central figure in helping to grow the ATP product across the globe. Our thoughts are with him and his family during this difficult time."
As a player, Drewett reached a career-high ranking of 34 in 1984 and was ranked in the top 20 in doubles, before moving into management with the ATP.